Chapter 9, Problem 18CR

To determine

To find the dimensions of a rectangle with perimeter $14$$m$ and diagonal is $5$$m$ .

Answer to Problem 18CR

The answer is $c.$

  $3$$m$ by $4$$m$ .

Explanation of Solution

Given:

Perimeter of rectangle $=$$14$$m$

Diagonal of rectangle $=$$5$$m$

Formula Used:

Perimeter of rectangle $=$$2\left(l+b\right)$

Diagonal of rectangle $=$$\sqrt{l^2+b^2}$

Calculation:

Given:

Perimeter of rectangle $=$$14$$m$

Diagonal of rectangle $=$$5$$m$

Using the above formula, we have:

Perimeter of rectangle $=$$2\left(l+b\right)$

  $\begin{array}{l}14=2\left(l+b\right)\\ l+b=7\\ l=7-b\end{array}$

And, Diagonal of rectangle $=$$\sqrt{l^2+b^2}$

  $5$

  $=$$\sqrt{b^2+{\left(7-b\right)}^2}$

  $\begin{array}{l}{b}^2+{\left(7-b\right)}^2=25\\ b^2+49+b^2-14b-25=0\\ 2b^2-14b+24=0\\ b^2-7b+12=0\\ b=4,3\end{array}$

Thus, $l=3,4$

Thus, length is $4$$m$ .

Breadth is $b=3$$m$ .

Thus, the length is $4$$m$ and breadth is $3$$m$ .

Conclusion:

The answer is $c.$

  $3$$m$ by $4$$m$ .